"Standing up there today and looking at the board and the stats that Cook is the youngest to put 7000 runs. It was quite an amazing moment! I was batting with this guy, the all-time leading century-maker for England," Compton couldn't stop praising his opening partner.
Compton says that Cook has a very clear head and knows what exactly he's doing.
"I would say he's very clear. He seems to know exactly what he's doing and what he wants to do. For me, it's easy to chill out there at the other end. It becomes a big difference when you have a player of Cook's calibre."
The grandson of legendary Dennis Compton complimented Cook for his fantastic temperament.
"Being very unflappable is probably a good way to describe him (Cook). He's tough. I've really enjoyed the time I spent out there," Compton said.
"He's quite an easy guy to get used to. He keeps things very simple. It's just a good way to get to know someone by getting there in the middle. You can't be closer to someone really in the heat of the battle. I really enjoyed that."
Compton says that Cook is only 27 and his achievements speaks for him.
"He's still young. A lot of players get there at my age probably at 28-29 but he's still very young at 27. It speaks volumes about him."
The Somerset batsman compared Cook with burly former opener Marcus Trescothick.
"To bat with him in someways is similar to batting with (Marcus) Trescothick at Somerset. Clearly, they are different players but how clinical they are. They make very few mistakes on each delivery. Every ball is played in a similar fashion, he always sees the ball, when it's released."
Compton feels that Cook's ability judge a delivery is exceptional.
"He always knows what's he's going to do. He has forward defence, the sweep shot, he seems to do that time and time again. That's the mark of a serious player. He does the simple things for a long period of time. He's been very very clear batting at the other end."
Compton scored his maiden half-century from three Test in a 165-run opening partnership with Cook.
"He's very clear in his gameplan and confidence as well. It does give the other players confidence. He's so solid. You know he's not going to give his wicket way easily. If we can stay with him, you're doing job."
Cook hit a six off Ravichandran Ashwin and Compton said he has all the shots in his armoury.
"The six was originally not a mark of his game but he has probably taken his game to another level with the array of shots."